F28PL1 Programming Languages Lecture 14: Standard ML 4

Polymorphic list operations length of list base case: [] ==> 0 recursion case: (h::t) => 1 more than length of t - fun length [] = 0 | length (_::t) = 1+length t; > val length = fn : ‘a list -> int - length [“a”,”b”,”c”]; > 3 : int length [“a”,”b”,”c”] ==> 1+length [“b”,”c”] ==> 1+1+length [“c”] ==> length [] ==> ==> 3

Polymorphic list operations append lists e.g. append [1,2,3] [4,5,6] ==> [1,2,3,4,5,6] NB not [1,2,3]::[4,5,6] :: wants ‘a and ‘a list not ‘a list and ‘a list recurse on 1st list base case: [] ==> 2nd list recursion case: (h::t) ==> put h on front of appending t to 2nd list

Polymorphic list operations - fun append [] l2 = l2 | append (h1::t1) l2 = h1::append t1 l2; > val append = fn : ‘a list -> ‘a list -> ‘a list - append [“a”,”b”,”c”] [“d”,”e”,”f”]; > [“a”,”b”,”c”,”d”,”e”,”f”] : string list

Polymorphic list operations - append [“a”,”b”,”c”] [“d”,”e”,”f”] ==> “a”::append [”b”,”c”] [“d”,”e”,”f”] ==> “a”::”b”::append [“c”] [“d”,”e”,”f”] ==> “a”::”b”::”c”::append [] [“d”,”e”,”f”] ==> “a”::”b”::”c”::[“d”,”e”,”f”] ==> - infix append operator - > [1,2,3,4,5,6] : int list

Polymorphic list operations is value e1 in list? base case: [] ==> e1 not in list- false recursion case 1: (e2::t) ==> e1 = e2 - true recursion case 2: (e2::t) ==> e1 <> e2 - is e1 in t - fun member _ [] = false | member e1 (e2::t) = e1=e2 orelse member e1 t; > val member = fn : ‘’a -> ‘’a list -> bool

Polymorphic list operations - member 7 [1,9,7,4]; > true : bool member 7 [1,9,7,4] ==> member 7 [9,7,4] ==> member 7 [7,4] ==> true

Polymorphic list operations add value e1 to list if not in list already base case: [] ==> make new list for e recursion case 1: (e2::t) ==> e1 = e2 – return (e2::t) recursion case 2: (e2::t) ==> e1 <> e2 – put e2 back after adding e1 to t will place new value at end of list

Polymorphic list operations - fun add e [] = [e] | add e1 (e2::t) = if e1=e2 then e2::t else e2::add e1 t; > val add = fn : ‘’a -> ‘’a list -> ‘’a list

Polymorphic list operations - add 1 [2,5,4]; > [2,5,4,1] : int list add 1 [2,5,4] ==> 2::add 1 [5,4] ==> 2::5::add 1 [4] ==> 2::5::4::add 1[] ==> 2::5::4::[1] ==> [2,5,4,1] - add 4 [2,5,4,1]; > [2,5,4,1] : int list add 4 [2,5,4,1] ==> 2::add 4 [5,4,1] ==> 2::5::add 4 [4,1] ==> 2::5::[4,1] ==> [2,5,4,1]

Polymorphic list operations delete value e1 from list base case: [] ==> can’t find e1 so return empty list recursion case 1: (e2::t) ==> e1 = e2 – return t recursion case 2: (e2::t) ==> e1 <> e2 – put e2 back after deleting e1 from t

Polymorphic list operations - fun delete _ [] = [] | delete e1 (e2::t) = if e1=e2 then t else e2::delete e1 t; > val delete = fn : ‘’a -> ‘’a list -> ‘’a list

Polymorphic list operations -delete “c” [“a”,”b”,”c”,”d”]; > [“a”,”b”,”d”] : string list delete “c” [“a”,”b”,”c”,”d”] ==> “a”::delete “c” [“b”,”c”,”d”] ==> “a”::”b”::delete “c” [“c”,”d”] ==> “a”::”b”::[“d”] ==> [“a”,”b”,”d”];

Higher order functions function which: 1.takes another function as parameter or: 2.returns a function as result natural in functional languages high degrees of: – abstraction – reuse polymorphic

Higher order function: filter often want to select those elements of a list for which some property holds filter list with predicate - fun filter _ [] = [] | filter p (h::t) = if p h then h::filter p t else filter p t; > val filter = fn : (‘a -> bool) -> ‘a list -> ‘a list if p holds for h then keep it

Higher order function: filter - fun filter _ [] = [] | filter p (h::t) = if p h then h::filter p t else filter p t; > val filter = fn : (‘a -> bool) -> ‘a list -> ‘a list p may be any ‘a -> bool function (h::t) must be ‘a list result must be ‘a list

Higher order function: filter e.g. find all in a list of integer > 0 - fun isPos x = x>0; > val isPos = fn : int -> bool - filter isPos [~2,1,0,2]; > [1,2] : int list filter isPos [~2,1,0,2] ==> filter isPos [1,0,2] ==> 1::filter isPos [0,2] ==> 1::filter isPos [2] ==> 1::2::filter isPos [] ==> 1::2::[] ==> [1,2]

Higher order function: map often want to create a new list by doing the same thing to each element of an old list map function over list - fun map _ [] = [] | map f (h::t) = f h::map f t; > val map = fn : (‘a -> ‘b) -> ‘a list -> ‘b list f may be any ‘a -> ‘b function (h::t) must be a ‘a list result must be a ‘b list

Higher order function: map e.g. find list of sizes for string list - map size [“a”,”bc”,”def”]; > [1,2,3] : int list size: string -> int f : ‘a -> ‘b so: ‘a == string; ‘b == int

Higher order function: map map size [“a”,”bc”,”def”] ==> size “a”::map size [“bc”,”def”] ==> size “a”::size “bc”::map size [“def”] ==> size “a”::size ”bc”::size “def”::map size [] ==> size “a”::size ”bc”::size “def”::[] ==> [1,2,3]

Higher order function: map e.g. find list of squares and cubes from integer list - fun powers (x:int) = (x,x*x,x*x*x); > val powers = fn : int -> int * int * int - map powers [1,2,3]; > [(1,1,1),(2,4,8),(3,9,27) : (int * int * int) list powers: int -> int * int * int f: ‘a -> ‘b so: ‘a == int ; ‘b == int * int * int

Higher order function: map map powers [1,2,3] ==> powers 1::map powers [2,3] ==> powers 1::powers 2::map powers [3] ==> powers 1::powers 2::powers 3:: map powers [] ==> powers 1::powers 2::powers 3::[] ==> [(1,1,1),(2,4,8),(3,9,27)]

Insert to insert an integer i1 into an ordered integer sequence in ascending order of first element base case: [] ==> [i1] recursion case 1: (i2::t) ==> i1 < i2 – put i1 on front of (i2::t) recursion case 2: (i2::t ) ==> i1 >= i2 – put i2 on front of inserting i1 into t

Insert - fun insert(i:int) [] = [i] | insert i1 (i2::t) = if i1<i2 then i1::(i2,e2)::t else i2::insert i1 t; > fn : int -> int list -> int list - insert 7 [5,9]; > [5,7,9] :int list insert 7 [5,9] ==> 5::insert 7 [9] ==> 5::7::[9] ==> [5,7,9]

Sort to sort a list of integers – insert head into sorted tail base case: [] ==> [] recursion case: (h::t) ==> insert h into sorted t - fun sort [] = [] | sort (h::t) = insert h (sort t); > fn : int list -> int list sort [7,9,5]; > [5,7,9] : int list

Sort sort [7,9,5] ==> insert 7 (sort [9,5]) ==> insert 7 (insert 9 (sort [5])) ==> i insert 7 (insert 9 (insert 5 (sort []))) ==> insert 7 (insert 9 (insert 5 []) ) ==> insert 7 (insert 9 [5])) ==> insert 7 [5,9] ==> [5,7,9]

Higher order function: foldr consider adding all elements of a list together: - fun sum [] = 0 | sum (h::t) = h+sum t; > val sum = fn: int list -> int - sum [1,2,3]; > 6 : int sum [1,2,3] ==> 1+sum [2,3] ==> 1+(2+sum [3]) ==> 1+(2+(3+sum [])) ==> 1+(2+(3+0)) ==> like doing + between elements of list

Higher order function: foldr consider doing f between elements of list fold base case: [] ==> return some base value b recursion case: (h::t) ==> apply f to h and result of folding f over t - fun foldr f b [] = b | foldr f b (h::t) = f h (foldr f b t); > val foldr = fn: (‘a->’b->’b) -> ‘b -> ‘a list -> ’b

Higher order function: foldr e.g use foldr to join all elements of string list together fun sJoin s1 s2 = s1^s2; val sJoin = string -> string -> string; foldr sJoin “” [“a”,”bc”,”def”]; “abcdef” : string

Higher order function: foldr foldr sJoin “” [“a”,”bc”,”def”] ==> sJoin “a” (foldr sJoin “” [“bc”,”def2]) ==> sjoin “a” (sJoin “bc” (foldr sJoin “” [“def”])) ==> sJoin “a” (sJoin “bc” (sJoin “def” (foldr sJoin “” []))) ==> sJoin “a” (sJoin “bc” (sJoin “def” “”)) ==> “abcdef”

Higher order function: foldr use foldr to make sum - fun add (x:int) y = x+y; > val add = fn: int -> int -> int do add between elements of list when list empty, return 0 -val sum = foldr add 0; > val sum = fn : int list -> int sum is like foldr with f == add and b == 0

Higher order function: foldr - sum [1,2,3]; > 6 : int sum [1,2,3] ==> foldr add 0 [1,2,3] ==> add 1 (foldr add 0 [2,3]) ==> add 1 (add 2 (foldr add 0 [3])) ==> add 1 (add 2 (add 3 (foldr add 0 []))) ==> add 1 (add 2 (add 3 0)) ==> 6

Higher order function: foldr use foldr to make sort do insert in between elements of list when list empty, return [] - val sort = foldr insert []; > val sort = fn : ‘a list -> ‘a list - sort [3,2,1]; > [1,2,3] : int list

Higher order function: foldr sort [3,2,1] ==> foldr insert [] [3,2,1] ==> insert 3 (foldr insert [] [2,1]) ==> insert 3 (insert 2 (foldr insert [] [1])) ==> insert 3 (insert 2 (insert 1 (foldr insert [] []))) ==> insert 3 (insert 2 (insert 1 [])) ==> [1,2,3]

Higher order insert generalise insert to work with list of arbitrary type - fun gInsert p v [] = [v] | gInsert p v (h::t) = if p v h then v::h::t else h::gInsert p v t > val gInsert = fn : (‘a ->’a->bool)-> ’a -> ’a list -> ‘a list if p holds between v and h then put v on front of list otherwise put h on front of inserting v into t with p

Higher order insert - fun iLess (x:int) y = x<y; > val iLess = fn : int -> int -> bool - val insert = gInsert iLess; > val insert = fn : int -> int list -> int list insert is like gInsert with p set to iLess

Higher order sort - fun gSort p [] = [] | gSort p (h::t) = gInsert p h (gSort p t); > val gSort = fn : (‘a -> ‘a -> bool) -> ‘a list ->’a list to sort a list with p, insert h with p into sorting t with p - val sort = gSort iLess; > val sort = fn : int list -> intlist sort is like gSort with p set to iLess

Higher order sort - fun gSort p = foldr (gInsert p) []; > val gSort = fn : (‘a -> ‘a -> bool) -> ‘a -> ‘a list -> ‘a list sorting with p is like folding with inserting with p